NAME:______________________

DATE:______________

HONORS/AP CALCULUS ———————————— THE RIEMANN SUM ———————————— A Riemann Sum is a mathematical construction that has a number of powerful real world applications. In fact you constructed Riemann Sums in some of your previous exercises. Here is how to construct a Riemann Sum. a) You must be given a function fx in the form of a data table, a continuous graph of a function without being given its equation or an equation of a function that can be graphed. b) You must be given a closed interval a, b, where a  b, along the domain of fx. c) You must divide a, b into a number of subintervals where the subintervals are not necessarily the same length though they can be and often are. The length of a sub interval is symbolized by Δx n where n is a positive integer and represent the particular subinterval. i) Therefore Δx 1 represents the length of the first subinterval;Δx 2 represents the length of the second subinterval;Δx 3 represents the length of the third subinterval;Δx 4 represents the length of the fourth subinterval; etc.... d) You must choose an Evaluation Point(EP), from the domain, in each subinterval. The EP can be at the endpoints of a subinterval, the midpoint of the subinterval or within the subinterval. i) If you choose the EP at the beginning of the subinterval, the EP is called a Left Hand Evaluation Point(LHEP). ii) If you choose the EP at the end of the subinterval, the EP is called a Right Hand Evaluation Point(LHEP). iii) If you choose the EP in the middle of the subinterval, the EP is called a Mid-Point Evaluation Point(MPEP).

1

iv) If you choose the EP within the given subinterval there is no special name for this type of EP. The symbol x n , where n is a positive integer, represents the EP in a subinterval. ...x 1 is used to symbolize the EP in the first subinterval. ...x 2 is used to symbolize the EP in the first subinterval. ...x 3 is used to symbolize the EP in the first subinterval. ...so on and so forth e) You then construct a SUM of PRODUCTS where each TERM in the sum is product of the length of the subinterval and the function evaluated at the EP in the respective subinterval. ABSTRACT MATH FORMULA FOR CONSTRUCTING A RIEMANN SUM Δx 1  fx 1   Δx 2  fx 2   Δx 3  fx 3   Δx 4  fx 4  . . . . Δx n  fx n  where n is a positive integer and represents the last subinterval ...If you used a LHEP in each subinterval you have constructed a Left Hand Riemann Sum. ...If you used a RHEP in each subinterval you have constructed a Right Hand Riemann Sum. ...If you used a MPEP in each subinterval you have constructed a Mid Point Riemann Sum. f) If you continue to construct Riemann Sums with more and more subintervals, i.e. the number of subintervals approach , you will find that your sum will start approaching a specific number . You then symbolize this number  with the following symbol: b

b

 a fxdx  

...The symbol  fxdx is called a "Definite Integral." a

... is called an "Integral Symbol." ...a is called the "lower limit." ...b is called the "upper limit." ...fx is called the "Integrand." ...dx is called the "differential of x" assuming the function is a function in terms of x.

2

SUMMING UP: I will assume that you will be able to apply a Riemann Sum to two specific applications in Chapter 3. These applications are a) Getting an accumulated quantity from a rate time graph over a given interval of time. b) Determining the area between a function and the horizontal axis over a given closed interval. ————————————————— EXAMPLES ————————————————— EXAMPLE: in 3 , at which air was entering the The function Vt  100 represents the rate, sec t1 balloon at any instant in time and t represents time in seconds. Vt models this real world situation over the interval of time 0, 8 or 0 ≤ t ≤ 8. a) Construct a Left Hand Riemann Sum of four equal subintervals in the closed interval 0, 8 to estimate how must air was blown into the balloon over the 8 second interval. Show units on your Riemann Sum ans: - I must divide 0, 8 into 4 equal subintervals of length 2 units. So each subinterval represents 2 seconds of time. The subintervals are as follows: Δx 1  0, 2 is 2 seconds Δx 2  2, 4 is 2 seconds Δx 3  4, 7 is 2 seconds Δx 4  6, 8 is 2 seconds - I must now choose an Evaluation Point in each subinterval. Since I want to do a Left Hand Riemann Sum I must choose the Left Hand Evaluation Point from each subinterval: x 1  0 sec is the EP in 0, 2 x 2  2 sec is the EP in 2, 4 x 3  4 sec is the EP in 4, 6 x 4  6 sec is the EP in 6, 8 - I must evaluate each LHEP in the given function: in 3 V0  100  100 sec 01 in 3 V2  100  100 sec 21 3 100 in 3 V4   20 sec 41

3

in 3 V6  100  100 sec 7 61

4

- I must construct a sum of products where each term is product of the length of the subinterval and the function evaluated at the EP in the respective subinterval. in 3   2 sec 100 in 3  2 sec 20 in 3  2 sec 100 in 3 2 sec100 sec sec 7 sec 3 sec 3 in of air entered the balloon. Approximately 7040 sec 21 Here is a visual representation of the above computations:

Teacher comment: Please note the symbol, ≈ , which means "approximately." 8 Please keep in mind that  100 dt represents a number equal to the numerical value 0 t1 that is approached if you used an infinite number of subintervals in your Riemann Sum. The value you got above is an ESTIMATION of a Riemann Sum with an infinite number of subintervals. 8 in 3  0 100 dt ≈ 7040 sec 21 t1

5

———————————————– USING fnInt ON YOUR TI ———————————————– Fortunately your TI has a built-in feature to get the exact answer to a definite integral, an infinite number of subinterval Riemann Sum. You have an option on your TI called fnInt which can get you the exact answer of a definite integral. The fnInt feature can be found by pressing the MATH button and choosing Option 9. I’ll show you how to use fnInt with the above problem. Step 1) Press MATH then go to Option 9 Step 2) You will see this Then press ENTER

Step 3) Enter in the following

Step 4) Press ENTER

0 is the beginning of your interval 8 is the end of your interval.

Your calculator has just done a Riemann Sum with an Infinite Number of Subintervals and has given you the result. Here is how you would write the answer. 8  0 100 dt  219. 722 457 7 in 3 of air

t1

As you can see using a definite integral is much easier than doing an Infinite Riemann Sum. BUT, you must be able to manually do a Riemann Sum of a finite number of subintervals to PROVE to me that you UNDERSTAND the amazing concept of a Riemann Sum.

6

————————————————————————————THE FUNDAMENTAL THEOREM OF CALCULUS(FTOC) ————————————————————————————– Assume that At  −10t 2  70t  70 models the amount of oil leaked from a tanker into Casco Bay where t is in hours and At is in cubic yards(yd 3 ) over the time interval 0, 7 hours. This tanker also has the ability to suck back the leaked oil into its tank. Teacher Comment: Please note that this is a QUANTITY TIME formula. Here is the graph.

Now lets apply the definition of the derivative to A(t) which will give us the IROC formula which will tell us HOW FAST oil is leaking into Casco Bay at any instant in time over the 7 second time interval. A ′ t  lim h→0

At  h − At h

At  h  −10t  h 2  70t  h  70  −10h 2  2ht  t 2   70t  h  70  −10h 2 − 20ht − 10t 2  70h  70t  70 At  −10t 2  70t  70 −10h 2 − 20ht − 10t 2  70h  70t  70 − −10t 2  70t  70 h 2 h70 − 20t − 10h A ′ t  lim h→0 70h − 20ht − 10h  lim h→0  lim h→0 70 − 20t − 10h h h 2 A ′ t  −20t  70 in hour A ′ t  lim h→0

Remember that A ′ t is the IROC formula. Let’s graph the IROC. A ′ t is telling you the rate at which the oil is entering and LEAVING Casco Bay.

7

There is a VERY IMPORTANT relationship between a quantity time graph and its derivative(a rate time graph). That relationship is clarified by the FUNDAMENTAL THEOREM OF CALCULUS(FTOC) whose proof is beyond the scope of this course. The FTOC can be stated as follows: b

fb − fa   f ′ xdx a

Now FTOC will be rewritten using the information above. t2

At 2  − At 1    A ′ xdx where t 2  t 1 t1

So what does the FTOC show you? Put on your thinking cap! Lets choose the times t  3 seconds and t  0 seconds. Lets construct the FTOC. 3 A3 − A0   −20t  70dt 0

Let’s compute A3 − A0 first. A3  −103 2  703  70  190 yd 3 of oil A0  −100 2  700  70  70 yd 3 of oil A3 − A0  190 yd 3 − 70 yd 3  120 yd 3 So 120 yd 3 is amount of oil that entered Casco Bay from t  0 hours to t  3 hours since we are using the function At which is the quantity formula.

8

3

Now lets compute  −20t  70dx using fnInt and the rate formula. Here is what I get. 0

3

 0 −20t  70dt  120 yd 3 So it appears that the FTOC holds true. Now lets look at the visual connection between the two graphs At and A ′ t. THINK! THINK! THINK!

7

What do you think A7 − A0   −20t  70dt, an example of the FTOC, would 0 compute to? And what would the results mean for Casco Bay and the Tanker? TRY IT AND PRODUCE A VISUAL REPRESENTATION AS I DID ABOVE.

9